Langmuir Isotherm Calculator
Use this calculator to apply the Langmuir adsorption isotherm to any liquid-phase or gas-phase system. Choose which parameter to solve for, enter the known values, and get the adsorbed amount, equilibrium concentration, maximum monolayer capacity, or Langmuir binding constant instantly. The results panel also shows the separation factor RL, the linearized Ce/q value, and a full isotherm curve so you can see where your point falls on the saturation plateau.
Formula
Worked example
With qmax = 10 mg/g, KL = 0.15 L/mg, and Ce = 20 mg/L: numerator = 10 * 0.15 * 20 = 30; denominator = 1 + 0.15 * 20 = 4; q = 30/4 = 7.5 mg/g. At C0 = 50 mg/L: RL = 1/(1+0.15*50) = 1/8.5 = 0.118 (favorable). Saturation = 7.5/10 = 75%.
What is the Langmuir isotherm?
The Langmuir isotherm, proposed by Irving Langmuir in 1916, describes how a solute (adsorbate) from a solution binds to a solid surface (adsorbent) at constant temperature. It rests on four assumptions: adsorption occurs on a fixed number of equivalent surface sites; each site holds at most one adsorbate molecule (monolayer coverage); all sites have identical adsorption energy; and adsorbed molecules do not interact with each other. When these assumptions hold, the relationship between the equilibrium concentration Ce and the amount adsorbed q follows a hyperbolic curve that levels off at a maximum capacity qmax. The Langmuir model is widely applied in water treatment (removal of heavy metals, dyes and pharmaceuticals), food science, drug delivery, catalysis and soil science.
How to use this calculator
Select which variable you want to solve for from the "Solve for" drop-down. The remaining fields become your inputs. For the most common use case - predicting how much of a contaminant an adsorbent removes - keep "Adsorbed amount (q)" selected, enter the maximum capacity qmax and Langmuir constant KL from a published or fitted isotherm, and type in the equilibrium concentration Ce you expect in your system. The calculator instantly returns q, the saturation fraction, the separation factor RL, and the linearized Ce/q value. Provide C0 (initial concentration) to get RL. Toggle on mass-balance mode to cross-check q from an experimental C0, Ce, volume and adsorbent mass - a mismatch signals a data or unit inconsistency. The isotherm chart shows the full curve with your operating point marked so you can see whether you are on the steep rising part or the flat saturation plateau.
The linearized form and how to fit isotherm parameters
Rearranging the Langmuir equation gives Ce/q = Ce/qmax + 1/(qmax*KL). If you plot Ce/q on the y-axis against Ce on the x-axis using a set of experimental data points (each pair from a different initial concentration), you get a straight line. The slope equals 1/qmax and the y-intercept equals 1/(qmax*KL), so you can read off both parameters directly from a linear regression. This calculator shows you the Ce/q value for your current inputs so you can build that linearization data set one point at a time. For robust fitting, collect at least five Ce-q pairs spanning from low to near-saturation conditions.
Separation factor RL and what it means
The dimensionless separation factor RL = 1/(1 + KL*C0) is calculated from the initial concentration C0 and the Langmuir constant KL. It tells you at a glance whether the isotherm is favorable. An RL between 0 and 1 means adsorption is favorable (the more you add, the more binds, but with diminishing returns - the typical Langmuir shape). RL equal to 1 reduces to a linear Henry isotherm (only valid at very low loadings). RL above 1 is unfavorable, meaning binding actually gets worse as concentration rises. RL equal to 0 implies completely irreversible binding. Favorable adsorption (low RL) is desirable in water treatment and fixed-bed column design because it leads to sharp adsorption fronts and high bed utilization.
Separation factor RL interpretation
| RL value | Adsorption type | Favorability |
|---|---|---|
| RL = 0 | Irreversible | Maximum removal |
| 0 < RL < 1 | Favorable | Good for separation |
| RL = 1 | Linear | Henry-type (low loading) |
| RL > 1 | Unfavorable | Poor removal efficiency |
RL = 1 / (1 + KL * C0) classifies the favorability of a Langmuir adsorption process.
Frequently asked questions
What units should I use for KL, Ce and q?
The Langmuir equation is dimensionally consistent as long as your units cancel correctly. The most common convention in environmental engineering is: Ce in mg/L, q in mg/g, and KL in L/mg. Then qmax is also in mg/g. You can also use mmol/L and mmol/g (with KL in L/mmol) for gas-phase or molar data. Whichever system you choose, apply it consistently - mixing mg and mmol within a single calculation gives wrong results.
Why does q have to be less than qmax?
The Langmuir model assumes a fixed number of surface sites. Once every site is occupied, no more adsorbate can bind, so q approaches but never exceeds qmax. If your measured q is equal to or greater than your fitted qmax, the Langmuir model has broken down - possible causes include the model being a poor fit to your data, or using a qmax fitted at a different temperature.
How is the Langmuir model different from the Freundlich model?
The Langmuir model assumes monolayer adsorption on a homogeneous surface with a finite number of sites, producing a curve that saturates. The Freundlich model is empirical and assumes a heterogeneous surface with an exponential distribution of binding energies - it does not predict saturation and has no finite maximum. In practice, many real adsorbents are better described by the Freundlich model at intermediate concentrations but by Langmuir near saturation. Both models are often tested against the same dataset and the better statistical fit is chosen.
What is the mass-balance mode and when should I use it?
In a batch experiment you add a known mass m of adsorbent to a volume V of solution at initial concentration C0, then measure Ce at equilibrium. Mass balance says the adsorbate that left the solution must be on the adsorbent: q = (C0 - Ce)*V/m. Toggle mass-balance mode on and fill in C0, V and m - the calculator computes q by mass balance and compares it with the isotherm q. If the two agree (within measurement noise), your data are internally consistent. A large discrepancy suggests a dosing error, a concentration measurement error, or precipitation of the adsorbate from solution rather than adsorption.
What does a high KL value mean physically?
KL is the ratio of the adsorption rate constant to the desorption rate constant. A high KL means adsorption is much faster than desorption - the adsorbate sticks tightly to the surface. Practically, a high KL adsorbent removes contaminants efficiently even at very low equilibrium concentrations, which is important for meeting drinking-water standards for trace pollutants. A low KL means the adsorbate binds weakly and will release back into solution more readily when the feed concentration drops.